Phase diagram of the interacting partially directed self-avoiding walk attracted by a vertical wall

TL;DR

This work rigorously analyzes the phase diagram of the interacting partially directed self-avoiding walk (IPDSAW) in the presence of an attractive vertical wall. By leveraging a random-walk representation and a bead-decomposition framework, the authors establish a surface transition inside the collapsed phase and derive explicit expressions for the critical curve δ_c(β) along with the order of the transition. They provide sharp asymptotics for the partition function across desorbed, adsorbed, and critical regimes, identifying universal exponents and Wulff-shape modifications due to wall pinning. Local limit theorems and detailed tilting changes of measure underpin the precise asymptotics, while bead-uniqueness and bead-shape results illuminate the polymer's geometry near the wall. Altogether, the paper rigorously confirms physicists’ predictions about wall-induced surface transitions and bead geometry, advancing the mathematical understanding of polymer collapse with boundary interactions.

Abstract

In the present paper, we consider the interacting partially-directed self-avoiding walk (IPDSAW) attracted by a vertical wall. The IPDSAW was introduced by Zwanzig and Lauritzen (J. Chem. Phys., 1968) as a manner of investigating the collapse transition of a homopolymer dipped in a repulsive solvent. We prove in particular that a surface transition occurs inside the collapsed phase between (i) a regime where the attractive vertical wall does not influence the geometry of the polymer and (ii) a regime where the polymer is partially attached at the wall on a length that is comparable to its horizontal extension, modifying its asymptotic Wulff shape. The latter rigorously confirms the conjecture exposed by physicists in (Physica A: Stat. Mech. \\& App., 2002). We push the analysis even further by providing sharp asymptotics of the partition function inside the collapsed phase.

Figures (4)

• Figure 1: Picture of the trajectory $\ell\in \mathcal{L}_{15,54}$ whose vertical stretches are $(3,4,-5,2,-3,0,0,7,-4,2,2,0,-6,3,-2)$. The wall interaction is highlighted in red, and the self-interaction is represented by a dashed line.
• Figure 2: Phase diagram of the polymer pinned at the vertical wall. The three phases ${\mathcal{E}}$, ${\mathcal{C}}$ and ${\mathcal{G}}$ are separated by the purple ($\delta = f(\beta,0)$), red ($\delta = \beta$) and green ($\beta = \beta_c$) curves. The surface transition between the regimes ${\mathcal{D}} {\mathcal{C}}$ and ${\mathcal{A}} {\mathcal{C}}$ is indicated by the blue curve ($\delta = \delta_c(\beta)$), for which we have an explicit expression. A change of convexity happens for the Wulff shape at the black curve ($\delta = \check \delta(\beta)$). The bounded grey set ${\mathcal{C}} _{\rm bad}$, see \ref{['eq:defCgood']}, is where our method fails and has been computed numerically. The first coordinate of the rightmost point in ${\mathcal{C}} _{\rm bad}$, denoted by $\beta_*$, is smaller than $\pi /\sqrt{3}\approx 1.81$ (rigorously) and around 1.47 (numerically).
• Figure 3: Schematic picture of a concave and a convex globule, on the left ($\delta>\check{\delta}(\beta)$) and right ($\delta < \check{\delta}(\beta)$) respectively.
• Figure 4: Bead decomposition of a trajectory. The blue lines stand for the polymer configuration, the orange dashed lines stand for the attractive self-interaction, and the red lines stand for the edges pinned at the attractive wall. In this example, we can see four beads each delimited by black dashed lines. The sign of the initial stretch of the third bead is determined by the last stretch of the previous bead. Since the second and fourth beads start with horizontal stretches, the sign of their first non-zero stretch may be either positive or negative.

Theorems & Definitions (117)

• Proposition 2.1
• Definition 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• proof : Proof of Theorem \ref{['The One Bubble']}
• Proposition 2.7
• Definition 3.1
• Lemma 3.2